I will review local index theorems in Quillen’s form for the families of Cauchy-Riemann operators on compact, punctured and orbifold Riemann surfaces - quotients of the Lobachevsky plane by the action of Fuchsian groups of the first kind.

Synge Lecture Theatre

14:30

Nikita Nekrasov

Bootstrapping instantons with antiinstantons

In quantizing classical mechanical systems to get (the corrections, non-perturbative
in $\hbar$) the eigenvalues of the Hamiltonian one often sums over the classical trajectories as in localization formulas, but also takes into account the contributions of the "instanton -- anti-instanton gas": an ill-defined set of approximate solutions of equations of motion. In the lecture we attempt to alleviate some of the frustrations of this 40+ yrs old approach by making use of honest solutions of equations of motion of the complexified classical mechanical system. The examples will include algebraic integrable systems, from the abstract Hitchin systems to the well-studied anharmonic oscillator. These ideas originate in the Bethe/gauge correspondence.

Synge Lecture Theatre

15:45

Yuri Tschinkel

Geometry over Nonclosed Fields

I will discuss recent advances in the study of rationality and unirationality of algebraic varieties.